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Convergence of Newton’s method for sections on Riemannian manifolds. (English) Zbl 1228.90155
The author explores the local behavior of Newton’s method for sections on Riemannian manifolds. Under the assumption that the covariant derivatives of the sections satisfy one kind of Lipschitz condition with \(L\)-average, new estimates of the radii of convergence balls of Newton’s method and the radii of uniqueness balls of singular points of sections on Riemannian manifolds are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve the corresponding results due to J.-P. Dedieu, P. Priouret and G. Malajovich [IMA J. Numer. Anal. 23, No. 3, 395–419 (2003; Zbl 1047.65037)] as well as the ones by C. Li and J. Wang [Sci. China, Ser. A 48, No. 11, 1465–1478 (2005; Zbl 1116.53024)]. Applications to special cases, which include the Kantorovich condition and the \(\gamma\)-condition, as well as Smale’s \(\gamma\)-theory for sections on Riemannian manifolds, are provided, which consequently improve the corresponding results in [Dedieu et al., loc. cit.].

MSC:
90C53 Methods of quasi-Newton type
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